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Chapter 20: Problem 4

Classify the following equations, specifying the order and type (linear or non-linear): (a) \(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}-\frac{\mathrm{d} y}{\mathrm{~d} x}=x^{2}\) (b) \(\frac{\mathrm{d} y}{\mathrm{~d} t}+\cos y=0\)

Short Answer

Expert verified
Question: Classify the given equations in terms of order and type (linear or non-linear): (a) \(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}-\frac{\mathrm{d} y}{\mathrm{~d} x}=x^{2}\) (b) \(\frac{\mathrm{d} y}{\mathrm{~d} t}+\cos y=0\) Answer: (a) Linear second-order equation, (b) Non-linear first-order equation.

Step by step solution

01

(Classification of Equation (a))

To classify equation (a), \(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}-\frac{\mathrm{d} y}{\mathrm{~d} x}=x^{2}\), first identify the highest derivative, which in this case is the second derivative (\(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}\)). Therefore, the order of this equation is 2. Next, determine if the equation is linear or non-linear. This equation is linear since the dependent variable and its derivatives appear only to the first power and are not multiplied together. So, equation (a) is a linear second-order equation.
02

(Classification of Equation (b))

To classify equation (b), \(\frac{\mathrm{d} y}{\mathrm{~d} t}+\cos y=0\), first identify the highest derivative, which in this case is the first derivative (\(\frac{\mathrm{d} y}{\mathrm{~d} t}\)). Therefore, the order of this equation is 1. Next, determine if the equation is linear or non-linear. This equation is non-linear because of the presence of the term \(\cos y\), which makes the equation not fit the linear equation description. So, equation (b) is a non-linear first-order equation.

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Most popular questions from this chapter

Integrate twice the differential equation $$ \frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}=\frac{w}{2}\left(l x-x^{2}\right) $$ where \(w\) and \(l\) are constants, to find a general solution for \(y\).

Use Euler's method to find a numerical solution of \(\frac{\mathrm{dy}}{\mathrm{d} x}=\frac{x y}{x^{2}+2}\) subject to \(y(1)=3\). Take \(h=0.1\) and hence approximate \(y(1.5)\). Obtain the true solution using the method of separation of variables. Work throughout to six decimal places.

The charge, \(q\), on a capacitor in an \(L C R\) series circuit satisfies the second-order differential equation $$ L \frac{\mathrm{d}^{2} q}{\mathrm{~d} t^{2}}+R \frac{\mathrm{d} q}{\mathrm{~d} t}+\frac{1}{C} q=E $$ where \(L, R, C\) and \(E\) are constants. Show that if \(2 L=C R^{2}\) the general solution of this equation is $$ \begin{aligned} &q= \\ &\mathrm{e}^{-t /(C R)}\left(A \cos \frac{1}{C R} t+B \sin \frac{1}{C R} t\right)+C E \end{aligned} $$ If \(i=\frac{\mathrm{d} q}{\mathrm{~d} t}=0\) and \(q=0\) when \(t=0\) show that the current in the circuit is $$ i=\frac{2 E}{R} \mathrm{e}^{-t /(C R)} \sin \frac{1}{C R} t $$

Find a first-order equation satisfied by \(x=A \mathrm{e}^{-2 x}\),

Give one example each of a first-order linear equation, first-order non-linear equation, second-order linear equation, second-order non-linear equation.
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